We find the critical time in the effective Hamiltonian is smaller than that in the Clifford simulation.
We provide a heuristic explanation here.
First, recall the effective Hamiltonian describes the transition in an approximate two-replica quantity, while the Clifford simulation estimates the properly averaged von Neumann entropy.
Furthermore, a \(d\)  dimensional finite-depth circuit with measurements at the final time maps to a \(d-1\)  dimensional hybrid quantum circuit, where measurement rate is inversely proportional to the evolution time [1]}.
We may view the all-to-all coupled circuit as short-range circuit in infinite dimension, which can itself be viewed as a local hybrid quantum circuit.
As shown in Ref. [2]}, [3]}, the critical measurement rate in the hybrid circuits increases as a function of replica index \(n\) .
Hence, we expect a lower critical time in higher replicas, consistent with our mean-field results.
